Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Differentiate $f (x) = {(3 x + \sqrt{x})}^{2}$ in three ways. When you have completed all three parts, show that your three answers are the same:
(a) with the chain rule
(b) with the product rule but not the chain rule
(c) without the chain or product rules.

The derivative of $f (x)$ using each rule is:localid="1648368040492" $f' (x) = 18 x + 9 \sqrt{x} + 1$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information:

The function is:

f (x) = {(3 x + \sqrt{x})}^{2}

Part (a). Step 1. To find derivative using chain rule:

$f (x) = {(3 x + \sqrt{x})}^{2}$

Let $u = 3 x + \sqrt{x}$

then $f (u) = u^{2}$

Derivative of $u$ with respect to $x$ :

$\frac{d u}{d x} = \frac{d}{d x} (3 x + \sqrt{x}) = 3 + \frac{1}{2 \sqrt{x}}$

Derivative of $f$ with respect to $u$ :

$\frac{d f}{d u} = \frac{d}{d u} (u^{2}) = 2 u = 2 (3 x + \sqrt{x})$

Derivative of $f$ with respect to $x$ :

localid="1648368008391" $\frac{d f}{d x} = \frac{d f}{d u} \times \frac{d u}{d x} = 2 (3 x + \sqrt{x}) (3 + \frac{1}{2 \sqrt{x}}) = 2 (9 x + \frac{3}{2} \sqrt{x} + 3 \sqrt{x} + \frac{1}{2}) = 2 (9 x + \frac{9}{2} \sqrt{x} + \frac{1}{2}) = 18 x + 9 \sqrt{x} + 1$

Part (b). Step 1. To find derivative using product rule:

$f (x) = {(3 x + \sqrt{x})}^{2} = (3 x + \sqrt{x}) . (3 x + \sqrt{x})$

$U = (3 x + \sqrt{x}) V = (3 x + \sqrt{x})$

localid="1648367311339" $\frac{d U}{d x} = \frac{d}{d x} (3 x + \sqrt{x}) = 3 + \frac{1}{2 \sqrt{x}} = \frac{d V}{d x}$

Now derivative of a function using the product rule is:

localid="1648367986645" $\frac{d f}{d x} = U \frac{d V}{d x} + V \frac{d U}{d x} = (3 x + \sqrt{x}) (3 + \frac{1}{2 \sqrt{x}}) + (3 x + \sqrt{x}) (3 + \frac{1}{2 \sqrt{x}}) = 2 (3 x + \sqrt{x}) (3 + \frac{1}{2 \sqrt{x}}) = 2 (9 x + \frac{3}{2} \sqrt{x} + 3 \sqrt{x} + \frac{1}{2}) = 2 (9 x + \frac{9}{2} \sqrt{x} + \frac{1}{2}) = 18 x + 9 \sqrt{x} + 1$

Part (c). Step 1. To find derivative without using chain rule or product rule:

$f (x) = {(3 x + \sqrt{x})}^{2} = {(3 x)}^{2} + {(\sqrt{x})}^{2} + 2 (3 x) (\sqrt{x}) = 9 x^{2} + x + 6 x^{\frac{3}{2}}$

Now differentiate this with respect to $x$ :

$\frac{d f}{d x} = \frac{d}{d x} (9 x^{2} + x + 6 x^{\frac{3}{2}}) = 9.2 x + 1 + 6 . \frac{3}{2} x^{\frac{1}{2}} = 18 x + 1 + 9 \sqrt{x}$

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Calculus

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Short Answer

Differentiate $f (x) = {(3 x + \sqrt{x})}^{2}$ in three ways. When you have completed all three parts, show that your three answers are the same:
(a) with the chain rule
(b) with the product rule but not the chain rule
(c) without the chain or product rules.

Step by Step Solution

Step 1. Given information:

Part (a). Step 1. To find derivative using chain rule:

Part (b). Step 1. To find derivative using product rule:

Part (c). Step 1. To find derivative without using chain rule or product rule:

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Calculus

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Short Answer

Differentiate f(x)=3x+x2in three ways. When you have completed all three parts, show that your three answers are the same: (a) with the chain rule (b) with the product rule but not the chain rule (c) without the chain or product rules.

Step by Step Solution

Step 1. Given information:

Part (a). Step 1. To find derivative using chain rule:

Part (b). Step 1. To find derivative using product rule:

Part (c). Step 1. To find derivative without using chain rule or product rule:

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Differentiate $f (x) = {(3 x + \sqrt{x})}^{2}$ in three ways. When you have completed all three parts, show that your three answers are the same:
(a) with the chain rule
(b) with the product rule but not the chain rule
(c) without the chain or product rules.